EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 61 



of these differential coefficients can have any value other than 0, for 

 the state of the system for which Srj e ^ 0. For otherwise, as it would 

 generally be possible, as before, by some infinitely small modification 

 of the case, to render impossible any change like or nearly like that 

 which might be supposed to occur, this infinitely small modification 

 of the case would make a finite difference in the value of the differ- 

 ential coefficients which had before the finite values, or in some of 

 lower orders, which is contrary to that continuity which we have 

 reason to expect. Such considerations seem to justify us in regarding 

 such a state as we are discussing as one of theoretical equilibrium; 

 although as the equilibrium is evidently unstable, it cannot be realized. 

 We have still to prove that the condition enunciated is in every 

 case necessary for equilibrium. It is evidently so in all cases in which 

 the active tendencies of the system are so balanced that changes of 

 every kind, except those excluded in the statement of the condition of 

 equilibrium, can take place reversibly, (i.e., both in the positive and 

 the negative direction,) in states of the system differing infinitely little 

 from the state in question. In this case, we may omit the sign of 

 inequality and write as the condition of such a state of equilibrium 



0, i.e., (<H = 0- (10) 



But to prove that the condition previously enunciated is in every 

 case necessary, it must be shown that whenever an isolated system 

 remains without change, if there is any infinitesimal variation in its 

 state, not involving a finite change of position of any (even an infini- 

 tesimal part) of its matter, which would diminish its energy by a 

 quantity which is not infinitely small relatively to the variations of 

 the quantities which determine the state of the system, without 

 altering its entropy, or, if the system has thermally isolated parts, 

 without altering the entropy of any such part, this variation involves 

 changes in the system which are prevented by its passive forces or 

 analogous resistances to change. Now, as the described variation in 

 the state of the system diminishes its energy without altering its 

 entropy, it must be regarded as theoretically possible to produce that 

 variation by some process, perhaps a very indirect one, so as to gain 

 a certain amount of work (above all expended on the system). Hence 

 we may conclude that the active forces or tendencies of the system 

 favor the variation in question, and that equilibrium cannot subsist 

 unless the variation is prevented by passive forces. 



The preceding considerations will suffice, it is believed, to establish 

 the validity of the criterion of equilibrium which has been given. 

 The criteria of stability may readily be deduced from that of equi- 

 librium. We will now proceed to apply these principles to systems 

 consisting of heterogeneous substances and deduce the special laws 



